The hypothesis (REF ) of the above proposition is effectively an instance of sure loss discussed in the imprecise probability literature; see, e.g., Condition (C7) in [1]} or Definition 3.3 in [2]}. Since sure loss is among the most egregious violations of rationality that the theory strives to avoid, we find comfort in the validity implies no sure loss conclusion that can be drawn from Proposition REF . Note that the exact same conclusion can be reached if the probabilistic predictor's lower probabilities are strictly and uniformly lower-bounded away from the marginal probability. This also reinforces our previous intuition that a probabilistic predictor cannot be valid unless it knows something about the true distribution; more on this point below.
